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Mertens' Third Theorem for Number Fields: A New Proof, Cramér's Inequality, Oscillations, and Bias

Shehzad Hathi, Ethan S. Lee

TL;DR

This work extends Mertens’ third product formula to number fields by introducing a new Hardy-style proof, examining oscillations of the associated error term, and studying bias via logarithmic densities and limiting distributions under GRH/GLI. It establishes a number-field analogue of Cramér’s inequality, proves sign changes of the error term in general and under GRH, and derives an explicit formula for the bias term with connections to the distribution of zeros of $\zeta_{\field{K}}(s)$. The paper also provides conditional results on the existence and form of limiting distributions, computes numerical densities for quadratic fields, and shows that bias dissipates as the discriminant grows. Together, these results deepen understanding of prime-ideal statistics, oscillations, and Chebyshev-type biases in algebraic number fields, with concrete numerical data for illustrative cases. The methods blend Hardy-style Tauberian arguments, explicit zero-sum formulas, and probabilistic models tied to zero distributions. The findings have implications for analytic number theory in number fields and for the broader study of sign-dynamics in prime-counting analogues.

Abstract

The first result of our article is another proof of Mertens' third theorem in the number field setting, which generalises a method of Hardy. The second result concerns the sign of the error term in Mertens' third theorem. Diamond and Pintz showed that the error term in the classical case changes sign infinitely often and in our article, we establish this result for number fields assuming a reasonable technical condition. In order to do so, we needed to prove Cramér's inequality for number fields, which is interesting in its own right. Lamzouri built upon Diamond and Pintz's work to prove the existence of the logarithmic density of the set of real numbers $x \ge 2$ such that the error term in Mertens' third theorem is positive, so the third result of our article generalises Lamzouri's results for number fields. We also include numerical investigations for the number fields $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{13})$, building upon similar work done by Rubinstein and Sarnak in the classical case.

Mertens' Third Theorem for Number Fields: A New Proof, Cramér's Inequality, Oscillations, and Bias

TL;DR

This work extends Mertens’ third product formula to number fields by introducing a new Hardy-style proof, examining oscillations of the associated error term, and studying bias via logarithmic densities and limiting distributions under GRH/GLI. It establishes a number-field analogue of Cramér’s inequality, proves sign changes of the error term in general and under GRH, and derives an explicit formula for the bias term with connections to the distribution of zeros of . The paper also provides conditional results on the existence and form of limiting distributions, computes numerical densities for quadratic fields, and shows that bias dissipates as the discriminant grows. Together, these results deepen understanding of prime-ideal statistics, oscillations, and Chebyshev-type biases in algebraic number fields, with concrete numerical data for illustrative cases. The methods blend Hardy-style Tauberian arguments, explicit zero-sum formulas, and probabilistic models tied to zero distributions. The findings have implications for analytic number theory in number fields and for the broader study of sign-dynamics in prime-counting analogues.

Abstract

The first result of our article is another proof of Mertens' third theorem in the number field setting, which generalises a method of Hardy. The second result concerns the sign of the error term in Mertens' third theorem. Diamond and Pintz showed that the error term in the classical case changes sign infinitely often and in our article, we establish this result for number fields assuming a reasonable technical condition. In order to do so, we needed to prove Cramér's inequality for number fields, which is interesting in its own right. Lamzouri built upon Diamond and Pintz's work to prove the existence of the logarithmic density of the set of real numbers such that the error term in Mertens' third theorem is positive, so the third result of our article generalises Lamzouri's results for number fields. We also include numerical investigations for the number fields and , building upon similar work done by Rubinstein and Sarnak in the classical case.
Paper Structure (19 sections, 21 theorems, 129 equations, 1 table)

This paper contains 19 sections, 21 theorems, 129 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. The Dedekind zeta-function
  4. The prime ideal theorem
  5. Cramér's inequality for number fields
  6. Littlewood's result for number fields
  7. Mertens' Product Formula for Number Fields
  8. Ingredients
  9. Proof
  10. Oscillations in Mertens' Third Theorem
  11. Proof of Lemma \ref{['lem:reinterpretation']}
  12. The non-GRH case
  13. The GRH case
  14. Bias in Mertens' Third Theorem
  15. Explicit formula
  16. ...and 4 more sections

Key Result

Theorem 1

If there exists a non-real zero $\sigma_\mathbb{K}$ of $\zeta_\mathbb{K}(s)$ such that $1/2 \le \operatorname{Re}(\sigma_\mathbb{K}) < 1$ and there is no zero in the right-half plane $\operatorname{Re}(s) > \operatorname{Re}(\sigma_\mathbb{K})$, then the quantity attains arbitrarily large positive and negative values as $x\to\infty$.

Theorems & Definitions (37)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Remark 6
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 27 more